Bayesian penalized Buckley-James method for high dimensional bivariate censored regression models

Abstract

For high dimensional gene expression data, one important goal is to identify a small number of genes that are associated with progression of the disease or survival of the patients. In this paper, we consider the problem of variable selection for multivariate survival data. We propose an estimation procedure for high dimensional accelerated failure time (AFT) models with bivariate censored data. The method extends the Buckley-James method by minimizing a penalized $L_2$ loss function with a penalty function induced from a bivariate spike-and-slab prior specification. In the proposed algorithm, censored observations are imputed using the Kaplan-Meier estimator, which avoids a parametric assumption on the error terms. Our empirical studies demonstrate that the proposed method provides better performance compared to the alternative procedures designed for univariate survival data regardless of whether the true events are correlated or not, and conceptualizes a formal way of handling bivariate survival data for AFT models. Findings from the analysis of a myeloma clinical trial using the proposed method are also presented.

Appendices

Multicollinearity design

Let sample size n be 100 and dimension p be 100. Let the first 10 variables be independently generated from standard normal distribution. Then for $j = 11,\cdots ,20$, consider

$$\begin{aligned} {\varvec{X}}_j = {\varvec{X}}_{j-10} + \tau , \end{aligned}$$

where $\tau$ is a random error from a standard normal distribution. The rest of the variables are further generated from multivariate normal distribution with mean zero and covariance matrix with elements $\varvec{\varSigma }_{ij} = 0.5^{|i-j|}$. Following Sect. 4.1, generate ${\varvec{T}}_{\cdot 1}$ and ${\varvec{T}}_{\cdot 2}$ and corresponding censoring times and censoring indicators. Furthermore, we assume the relevant variables as the following

• no sharing: $\left\{ j: \varvec{\beta }_{j1} \ne 0\right\} = \left\{ 1,\cdots ,10\right\} , \left\{ j: \varvec{\beta }_{j2} \ne 0\right\} = \left\{ 21,\cdots ,30\right\}$

• all sharing: $\left\{ j: \varvec{\beta }_{jk} \ne 0\right\} = \left\{ 1,\cdots ,10\right\}$

• some sharing: $\left\{ j: \varvec{\beta }_{j1} \ne 0\right\} = \left\{ 1,\cdots ,10\right\} , \left\{ j: \varvec{\beta }_{j2} \ne 0\right\} = \left\{ 1,\cdots ,5,\cdots ,\right. \left. 21,\cdots ,25\right\}$

All of the true relevant variables are generated independently from $\textsf {N}(3,0.5)$. We repeat all simulation setups for 200 times and fix the true coefficient values for all simulation runs.

Table 11 False positives and false negatives reported for multicollinearity design

Table 12 Sensitivity, specificity, and MCC scores reported for multicollinearity design

The results of the multicollinearity design can be found in Tables 9 and 10. For this simulation design, the univariate AEnet failed to give any results due to the issue with singular matrix computation, therefore we only report results from the other four competing methods. We see that all of the methods tend to recognize the ten irrelevant variables as signals. For no-sharing and all-sharing cases, the proposed method is able to give the smallest number of false positives while being able to recognize almost all of the relevant variables, giving almost zero false negatives. For some-sharing cases, we observe more obvious trade-off between false positives and false negatives for using $\lambda _{min}$ and $\lambda _{1se}$ while the proposed method selects the variables more strictly, returning with lower false positive scores and higher false negative scores. However, in terms of MCC score as an overall measure, the proposed method is able to achieve the highest MCC scores for all setups, demonstrating that the proposed method is able to outperform existing methods and to handle complicated data examples.

Dense design

Let $n = 100$ and $p = 100$. Following Sect. 4.1, we generate design matrix ${\varvec{X}}$ from multivariate normal distribution with mean zero and covariance matrix with elements $\varvec{\varSigma }_{ij} = 0.5^{|i-j|}$. Then we generate ${\varvec{T}}_{\cdot 1}$ and ${\varvec{T}}_{\cdot 2}$ and corresponding censoring times and censoring indicators in a similar manner. In this simulation design, we assume that for each column of the true coefficient matrix, there are 20 relevant variables. That is, for some-sharing setups, we will have in total 45 relevant variables. All of the true relevant variables are generated independently from $\textsf {N}(3,0.5)$. We repeat all simulation setups for 200 times and fix the true coefficient values for all simulation runs.

Table 13 False positives and false negatives reported for dense design

Table 14 Sensitivity, specificity, and MCC scores reported for dense design

The results of the dense design can be found in Tables 11 and  12. We see that the proposed method gives consistent performance to have the best MCC scores among all competing methods. For no-sharing and all-sharing setups, the proposed method is able to give the best combination of false positives and false negatives, achieving highest sensitivity and specificity scores. For some-sharing setups, when $c \ne 1$, the proposed method is more strict in selecting signals which results in missing almost half of the relevant variables. However the proposed method is still able to correctly identify more relevant variables and noise variables compared with other competing methods, achieving the highest MCC scores.